In haddonm/datalowSA: A Package to Faciliate Application of Data poor Stock Assessments

knitr::opts_chunk$set(
    echo = TRUE,
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    warning = FALSE)
options(knitr.kable.NA = "",
        knitr.table.format = "pandoc")

options("show.signif.stars"=FALSE,"stringsAsFactors"=FALSE,
        "max.print"=50000,"width"=240)

library(datalowSA)
suppressPackageStartupMessages(library(knitr))

Introduction

Which Assessment to Apply

Which stock assessment method to apply to fisheries for data-poor to data-moderate species will depend upon what fisheries and biological data are available but also, importantly, on what management objectives need to be met within the jurisdiction in question. It may be the case that the fishery for a particular species is of sufficient size and value to warrant on-going monitoring and management towards some defined goal for the stock. In such a case the assessment used should obviously be capable of generating some notion of the current state of the fishery and indicate what management actions may be required to eventually achieve the agreed management goals. But some fisheries may be so minor that trying to actively manage them would be inefficient. Nevertheless, to meet the requirements of the Status of key Australian Fish Stocks (SAFS) one still requires some form of defensible stock assessment capable of determining whether the current level of fishing is sustainable.

Catch Curve Analyses

Catch curves are an assessment approach that has a relatively long history. The idea behind them is very simple. If the total population concerned received constant recruitment and a constant mortality rate each year, such an equilibrium assumption would imply that all cohorts would be identical and the numbers in each age class would be expected to decline exponentially. It is possible to follow the fate of a single cohort through a number of years, or consider multiple cohorts in a single year, or both of those options

$${{N}{t}}={{N}{0}}{{e}^{-(M+F)t}}={N}{0}{e}^{-Zt}={N}{t-1}{e}^{-Z}$$ where N~t~ is the numbers at a particular age (a cohort) t, M is the natural mortality, F is the age-specific fishing mortality, Z is the age-specific total mortality. This relationship can be linearized through log-transformation.

$$Log({N}{t})=Log({N}{0})-Zt$$ which transforms the exponential relationship into a simple linear regression. We can use this linear relationship to generate an estimate of total mortality. First we assume that fishing mortality and natural mortality have been constant (or at least have varied consistently), and that one has the age-composition of the catch in a single year. If one then plots the log-transformed numbers-at-age against age, the gradient of a linear regression through the data will provide an estimate of total mortality. This can be illustrated through the use of a particular data set and we will use one of the built in data sets: plaice.

# library(datalowSA)
data(plaice)
str(plaice,max.level=1)
naa <- plaice$agedata$naa
pick <- which(naa[,"sex"] == 1)
yrs <- naa[pick,"year"]
naa <- naa[pick,3:11]
ages <- plaice$agedata$ages
#plotprep()
compyear(naa,yrs,ages,plots=c(3,3),freq=TRUE,border=3)

Figure 1. Numbers-at-age for each year of data from the plaice data set.

When the age data from the plaice data set are considered it is clear that the notion of a exponential decline in numbers-at-age is only approximated across sequential ages by this real data. Such deviations from the theoretical expectation derive from the fact that real populations are not in equilibrium and recruitment is not a constant. For example, a strong year class becomes apparent in 1932, and the progression is relatively easily seen through 1933 and 1934. So the expected dynamics appears to be realistic although the assumptions about equilibrium do not appear to be valid. Despite this we can use this data to illustrate the use of the catch curves.

A standard workflow when using catch curves might consist of:

1. read in age composition data (and natural mortality)
2. visualize that age composition using the function compyear.
3. use naiveCC for the classical catch curve.
4. alternatively use selectCC where the selectivity is estimated as well as the gradient of decline.

Application of The Classical Catch Curves

One obvious aspect of the age composition data illustrated in Figure 1 is that the earlier ages classes are less common than later age classes. This is a reflection of the effects of selectivity on what gets caught. The classical catch curve primarily involves fitting a linear regression to the log-transformed numbers-at-age. Given the effect of selectivity it would be invalid to simply fit a regression to all age data. To avoid the effects of selectivity the classical catch curve requires one to select an age from which to fit the regression.

M <- plaice$glb$M
pickage <- 4  # try repeating this but compare with pickage = 3
numaa <- naa[1,]   # repeat this using different years of data
out <- classicCC(M,ages,numaa,pickage,plot=TRUE)

Figure 2. The classical catch curve applied to the age data from 1929, from the plaice data set. The age at which the regression line begins to be fitted in the plot was age 4 (pickage = 4). If a pickage of 3 was chosen the regression line would be somewhat somewhat less steep, which would have implied a lower fishing mortality rate.

If the classicCC is applied to each of the nine years of data it would be important to use the same pickage. In some cases (possibly 1932 - 1934) the strong year class can disrupt this analysis by suggesting that selectivity only operates from a much older pickage.

M <- plaice$glb$M
pickage <- 4   # try repeating this but compare with pickage = 4
numaa <- naa[5,]   # repeat this using different years of data
out <- classicCC(M,ages,numaa,pickage,plot=TRUE)

Figure 3. The classical catch curve applied to the age data from 1933, from the plaice data set. The age at which the regression line begins to be fitted in the plot was age 4 (pickage = 4). If considered alone this plot appears to suggest that a pickage of 5 might be chosen, but that would be merely a reflection of the strong year class rather than the original pickage of 4.

print(out)

By comparing Figures 2 and 3 it should be clear that only having a single year of catch age composition data would include a risk that the interpretation of the selectivity within the fishery concerned may be biased. The interpretation of catch curves becomes compromised by the presence of very strong year classes and their presence usually means use of catch curves becomes invalid.

Inclusion of Selectivity in Catch Curves

Rather than just ignoring the earlier weakly selected ages, as in the classical catch-curve, it is possible to estimate selectivity parameters as well as the total mortality rate. However, this requires the use of an equilibrium based age-structured model rather than the log-transformed linear regression (see equations in the Appendix below; Wayte, S.E. and N.L. Klaer, 2010).

data(plaice)
glb <- plaice$glb
naa <- plaice$agedata$naa
pick <- which(naa[,"sex"] == 1)
naa <- naa[pick,3:11] # use only the numeric data
numaa <- naa[2,]    # use the data from 1930
pars <- c(A50=3.0,delta=0.5,fcur=0.6)
out <- selectCC(glb$M,glb$maxage,numaa,pars,plot=TRUE)

Figure 4. This fittted catch curve uses a simple age-structured model to include the estimation of selectivity from the age data. It provides an estimate of fully selected fishing mortality rather than an average fishing mortality applied to all included age classes (as in the classicCC).

print(out$best)

kable(out$result,digits=c(0,3,3,3,4,4,5,4))

Discussion

Still to come.

Management Advice from Catch Curves

Still to come.

References

Dick, E.J. and A.D. MacCall (2011) Depletion-based stock reduction analysis: a catch-based method for determining sustainable yields for data-poor fish stocks. Fisheries Research 110(2): 331-341

Haddon, M. (2014) Tier 4 analyses in the SESSF, including deep water species. Data from 1986 – 2012. Pp 352 – 461 in Tuck, G.N. (ed) (2014) Stock Assessment for the Southern and Eastern Scalefish and Shark Fishery 2013. Part 2. Australian Fisheries Management Authority and CSIRO Marine and Atmospheric Research, Hobart. 313p.

Haddon, M., Klaer, N., Wayte, S., and G. Tuck (2015) Options for Tier 5 approaches in the SESSF and identification of when data support for harvest strategies are inappro-priate. CSIRO. FRDC Final Report 2013/200. Hobart. 115p.

Kimura, D.K. and J.V. Tagart (1982) Stock Reduction Analysis, another solution to the catch equations. Canadian Journal of Fisheries and Aquatic Sciences 39: 1467 - 1472.

Kimura, D.K., Balsiger, J.W., and Ito, D.H. 1984. Generalized stock reduction analysis. Canadian Journal of Fisheries and Aquatic Sciences 41: 1325–1333.

Little, L.R., Wayte, S.E., Tuck, G.N., Smith, A.D.M., Klaer, N., Haddon, M., Punt, A.E., Thomson, R., Day, J. and M. Fuller (2011) Development and evaluation of a cpue-based harvest control rule for the southern and eastern scalefish and shark fishery of Australia. ICES Journal of Marine Science 68(8): 1699-1705.

Martell, S. and R. Froese (2013) A simple method for estimating MSY from catch and resilience. Fish and Fisheries 14: 504-514

Punt, A.E., Butterworth, D.S. and A.J. Penney (1995) Stock assessment and risk analysis for the South Atlantic population of albacore Thunnus alalunga using an age-structured production model South African Journal of Marine Science 16: 287-310. http://dx.doi.org/10.2989/025776195784156476

R Core Team (2017). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/. see also https://cran.r-project.org/

RStudio (2016) www.rstudio.com

Schaefer, M.B. (1954) Some aspects of the dynamics of populations important to the management of the commercial marine fisheries. Bulletin, Inter-American Tropical Tuna Commission, 1: 25-56.

Schaefer, M.B. (1957) A study of the dynamics of the fishery for yellowfin tuna in the Eastern Tropical Pacific Ocean. Bulletin, Inter-American Tropical Tuna Commission, 2: 247-285

Walters, C.J., Martell, S.J.D. and J. Korman (2006) A stochastic approach to stock reduction analysis. Canadian Journal of Fisheries and Aquatic Sciences 63: 212 - 223.

Wayte, S.E. and N.L. Klaer (2010) An effective harvest strategy using improved catch-curves. Fisheries Research 106: 310-320.

Appendix: Catch Curve Equations

Catch-Curves with Selectivity

A simple model of the expect explonential decline of numbers-at-age is needed:

$$\begin{matrix} {N}{0}= & 1 & a=0 \ {N}{a}= & {{N}{a-1}}\exp \left( -{{s}{a-1}}F-M \right) & 0<a<{{a}{\max }} \ {N}{a}= & \frac{{{N}{a-1}}\exp \left( -{{s}{a-1}}F-M \right)}{\left( 1-\exp \left( -{{s}{a}}F-M \right) \right)} & a={{a}{\max }} \ \end{matrix}$$

where $N_a$ is the numbers-at-age $a$, $s_a$ is the selectivity-at-age $a$, $F$ is the fully selected average fishing mortality over the years for which ages are available, $M$ is the assumed natural mortality, and $a_{max}$ is the maximum age. The selectivity is assumed to be asymptotic (so this method may not be suitable for gillnets or hook gear):

$$s_a = \frac{1}{1+exp(-\ln(19)(a-a_{50})/\delta)}$$ where $a$ is age, $a_{50}$ is the age at 50% selectivity and $\delta$ is equal to $a_{95} - a_{50}$.

Three parameters are estimated, both the selectivity paramters, $a_{50}$ and $\delta$, and the average fully selected $F$ value. $\delta$ is used instead of $a_{95}$ as this reduces interactions between the two and helps avoid $a_{50}$ values larger than $a_{95}$ values. These parameters can be estimated through minimizing the multinomial negative log-likelihood:

$$-veLL=-\sum\limits_{y}\sum\limits_{a}{{O}{y,a}}log\left(\frac{{\hat{N}{a}}}{\sum{{\hat{N}_{a}}}} \right)$$

where $n_y$ is the sample size $n$ in year $y$, $O_{y,a}$ is the observed numbers-at-age $a$ in the catch in year $y$, and ${\hat{N}_{a}}$ is the expected or predicted numbers-at-age $a$ in the catch in year $y$. The predicted numbers-at-age are calculated using the Baranov catch equation:

$${\hat{N}_a}=\frac{{N_a}{s_a}F}{(s_aF+M)}\left(1-e^{-(s_aF+M)} \right )$$

haddonm/datalowSA documentation built on Nov. 5, 2023, 6:40 p.m.